Follow up: Omega got back to me and said to send my MK IV to the New Jersey repair center. I'll post how the experience went and before and after photos when it is done.

As a rule, vintage pieces would go to the main SGUS facility in Secaucus, and from their if the piece is still too old it'll get sent to Bienne/Biel (which requires a $75 shipping charge that will get rolled into your bill if you approve to send it for an estimate).